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STRESS CONCENTRATION AT NOTCHES One of the fundamental issues of designing a fatigue resistant structure (“design against fatigue”) is the consideration of stress concentration Stress concentration at geometrical notches are always present in a real structure Notches introduce inhomogeneous stress distribution with a stress concentration at the root of the notch Stress concentration factor: K t describes the severity of the notch and depends on the geometry of the notch configuration (shape factor of the notch) K t is referred as the theoretical stress concentration factor: it is based in the assumption of linear elastic material behavior

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Common examples of stress concentration (a)Gear teeth (b)Shaft keyway (c)Bolt threads (d)Shaft shoulder (e)Riveted or bolted joint (f)Welded joint all these components might be subjected to cyclic loads !

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DEFINITIONS: For the previous example: therefore: following the definitions of R.E. Peterson in Stress Concentration Factors, John Wiley & Sons, New York (1974) In general K t is the preferred factor to indicate stress concentration usualalternative

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THE “MODEL” STRESS CONCENTRATION CASE: the circular hole in an infinite sheet S S r r0r0 for = 0 ? Along the edge of the circular hole: compression at = 0 = 30° or = /2 – from max

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Stress Profiles along the normal to the edge of the circular hole: We are interested in evaluating: a) Situation for compressive remote stress (–S): presence of tensile stress ? YES b) Gradient of Stress in the direction normal to the edge of the hole at the location of s peak : strong gradient c) Gradient of Stress along the edge of the hole at the location of s peak : slow decrease of stress along the edge of the notch d) Volume of material subjected to high Stress around the root of the notch: larger for larger notches! (significant to understand notch size effects on fatigue)

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a)Presence of Local Tensile Stress upon Remote Compression Stress * Fatigue Crack Growth under Compressive Stress? * Effect on Brittle Materials: Failure Criteria Based on Maximum Normal Stress for Brittle Materials How would a cylinder of brittle material with a distribution of defects fail under compression? Spherical voids and sharp like cracks are often produced during processing of brittle materials: - Spherical voids sintering of ceramic powders (remnants of initial porosity) - Microcraks thermal expansion mismatch CC CC

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b) Gradient of Stress in the direction NORMAL to the edge of the hole at the location of peak * Although the peak stress is of great importance, it is also interesting to know how fast the stress decreases away from the root of the notch Stress Gradient for the circular hole: The Stress Gradient at the root of a notch should give an indication of the volume of material under high stresses estimate the distance along the normal to the root for a drop from peak to 0.9 peak (10 % decrease) for a circular hole with r 0 = 2.5 mm: = 0.1 mm= 100 m if grain size 50 m the depth corresponds only to few grains The grains at the noth root surface are subjected to high loads and this is very important for fatigue

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S S c) y d) Gradient of Stress ALONG the edge of the hole at the location of peak As fatigue crack nucleation is a surface phenomenon it is of interest to know how fast the tangential stress along the edge of the notch is decreasing * Slow decrease of the stress along the edge compared with the decrease from the edge at the location of peak * Larger notches have a larger material surface along the root of the notch very important to understand notch size effects on fatigue Note: even when a particular case was analyzed here (circular hole in an infinite plate) the conclusions are of general validity similar peak stresses and notch root radii give comparable stress distribution around the root of an arbitrary notch

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Geometrically similar specimens have the same K t (K t is dimensionless) … Larger specimens have larger volumes and larger notch surface areas of highly stressed material Reason of the existence of Notch Size Effects in Fatigue … but different stress gradients (stress gradient is not dimensionless) Also: Importance of surface quality (method of production / fabrication) surface defects due to manufacturing in a highly stressed region along the wall of a hole Effect of notch geometry on K t Further examples / further aspects of stress concentrators

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The elliptical hole in an infinite sheet a / b1/313 / a 911/9 KtKt 1.6737 S S use large radii on surface parallel to applied stress to reduce stress concentration !

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Stress Concentration for an elliptical hole under biaxial loading: * For the case of a thin walled pressure vessel under pressure = 0.5 and for the case of a circular hole (a = b): lower than 3 S for uniaxial loading * Same case but elliptical hole with b/a = 2: lower than 3 S for uniaxial loading (actually, = 1.5 S along the edge of the hole) compare with the square hole (dashed line) with rounded corners with r 10% of hole width: K t = 4.04 for = 0.5 : biaxiality ratio S

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Stress Concentration for a circular hole in a plate under pure shear: Fatigue cracks growing from holes in a shaft subjected to cyclic torsion !

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Pin - loaded hole: Comparison of K t values for a lug and an open hole * Lugs are fatigue critical parts (also prone to fretting corrosion) values of d/W below 1/3 are usually avoided to keep K t below 3.5 Conection between a lug and a clevis:

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Superposition of notches: If a relativelly small notch is added to the root of the main notch Effect of superposition of notches: This overestimates K t because the small notch is not completely embedded in an homogeneous stress field of magnitude K t1

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Technique for estimating conservative limiting value for K t for superposed notches: The theoretical stress concentration factor for the single deep narrow notch will always be greater than the K t for the multiple notch (see K t for Edge Notches two transparencies later) “Fill” the notch (cross hatched area) leaving a single deep narrow notch

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Examples of Superposition of notches: Cross section of a fatigue crack at a sharp corner Lug with small lubrication hole to the lug hole

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Edge notches and Corrosion Pits Corrosion pits at the material surface of an Al-alloy. Pit depth = 0.15 mm. Equivalent shape gives very high K t values

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Stress concentration factors for a shaft with a grove subjected to: Axial Load Torsion Bending Further information of the type that can be found in the clasical handbook of R.E. Peterson, Stress Concentration Factors, 1974

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S a = S e S eK THE FATIGUE STRENGHT OF NOTCHED SPECIMENS * STRESS – LIFE APPROACH: Notch Effects on the Fatigue Limit (S m = 0) Similarity Principle: if S a = S e is the fatigue limit of the smooth specimen, then S peak should give the fatigue limit S eK of the notched specimen: meaning that: … but this is not the case ! The Fatigue Strength Reduction Factor or Fatigue Notch Factor K f is introduced: In general: K f < K t fatigue limit of different materials are less notch sensitive to fatigue than predicted by K t

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Effect of a Notch on S - N behavior (Tryon and Day, 2003) Examples:

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Mechanical Behaviour of Materials (Dowling, 1999) The examples illustrates a general observation for different materials: the finite life region is also less notch sensitive to fatigue than predicted by K t

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In general K f < K t Blunting effects in soft materials: Yielding at the notch root reduces peak stress from the values predicted by K t Fatigue strength of a notched component depends on the volume of highly stressed material near the notch also effect of stress gradient on crack growth. K t depends on geometry and mode of loading K f also depends on material and notch size and

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For engineering applications, the fatigue strength reduction fator K f can be empirically related to the elastic stress concentration facto K t by a Notch Sensitivity Factor defined as: q = 1 material fully notch sensitive: K f = K t q = 0 material not notch sensitive: K f = 1 Empirical equations for q were proposed by different authors: * Peterson (1959) * Neuber (1946) * Siebel and Stiele (1955)

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* Peterson assumed that fatigue damage occurs when a the stress at a point located at a critical distance a p away from the notch root is equal to the fatigue strength of a smooth specimen and obtained the following empirical equation: - r is the notch root radius - a P is a material constant related with material strength and ductility. Effect of notch root radius on K f * for high strenght steels with S U > 560 MPa:

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Peterson ´s notch sensitivity for steels Also, q can be obtained in graphical form:

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* Neuber assumed that fatigue failure occurs if the average stress over a length a N from the notch root is equal to the fatigue limit of a smooth specimen and proposed the following empirical equation: - r is the notch root radius - a N is the Neuber´s material constant related to the grain size Neuber´s Notch Sensitivity curves for Al alloys Relation with Hall-Petch?

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where: for the circular hole: * Siebel and Stiele (1955) introduced the Relative Stress Gradient (RSG) to characterize the effects of fatigue strength reduction (instead of using the notch radius!) No significant effect of K t on Similar dependencies are found for other geometries and for typical K t values: 2 < K t <5

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Testing the Fatigue Strength of smooth and notched specimens they generated empirical curves relating K f / K t vs. Equation for the Curves: where C ss is a material constant related with S y

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RSG ( ) values calculated from Siebel and Stiele by the theory of elasticity for different notched members Examples:

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* STRESS – LIFE APPROACH: Notch Effect on Finite Life (S–N curve) In the Finite Life region the notch may be less sensitive to what is predicted by K f for the Fatigue Limit to much conservative ! Use K f for the whole S-N curve? : define a Fatigue Sensitivity Factor at a particular Life (and interpolate / extrapolate) analogously to what was made for estimating S – N curves for smooth specimens K f for N 1000 : K’ f

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An Empirical Fatigue Notch Sensitivity Factor (q’ 1000 ) can be defined at 1000 cycles:

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* Estimate of Fatigue Life for Notched Component: different approaches Fatigue Notch Sensitivity Factor K f depends on Cycles to Failure N f A) “Juvinall” approach B) Examples: LCF REGION!

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Example: Juvinall for the following loading cases: (1)Reversed bending loading (2)Reversed axial loading (negligible bending) (3) Reverse torsional loading

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